 # Log base 2

### What are Logarithmic Functions?

• A logarithmic function is a function that is the inverse of an exponential function.

• The purpose of the logarithm is to tell us about the exponent.

• Logarithmic functions are used to explore the properties of exponential functions and are used to solve various exponential equations.

 Representation of a Logarithm Function$lo{g_a}b{\text{ }} = x,{\text{ }}then{\text{ }}{a^x} = b$

### What is Log Base 2 or Binary Logarithm?

• Log base 2 is also known as binary logarithm.

• It is denoted as (log2n).

• Log base 2 or binary logarithm is the logarithm to the base 2.

• It is the inverse function for the power of two functions.

• Binary logarithm is the power to which the number 2 must be raised in order to obtain the value of n.

• Here’s the general form.

${\mathbf{x}}{\text{ }} = {\text{ }}{\mathbf{lo}}{{\mathbf{g}}_{\mathbf{2}}}{\mathbf{n}}\;\;\;\boxed{} - - - - - - - - - \boxed{}{{\mathbf{2}}^{\mathbf{x}}} = {\text{ }}{\mathbf{n}}$

### Properties of Log Base 2

There are a few properties of logarithm functions with base 2. They are listed in the table below.

Since we are discussing log base2, we will consider the base to be 2 here.

## Basic Log Rules

 Product Rule – If the logarithm is given as a product of two numerals, then we can represent the logarithm as the addition of the logarithm of each of the numerals and vice versa.$lo{g_b}\left( {x{\text{ }} \times {\text{ }}y} \right){\text{ }} = {\text{ }}lo{g_b}x{\text{ }} + {\text{ }}lo{g_b}y$ Quotient Rule –If the logarithm is given as a ratio of two quantities, then it can be written as the difference of the logarithm of each of the numerals.$lo{g_b}\left( {\frac{x}{y}} \right)\; = {\text{ }}lo{g_b}x{\text{ }} - {\text{ }}lo{g_b}y$ Power Rule -If the logarithm is given in exponential form, then it can be written as exponent times the logarithm of the base.$logb({x^k}) = k{\text{ }}lo{g_b}x$ Zero Rule –If b is greater than 0, but not equal to 1. The logarithm of x= 1 can be written as 0.$lo{g_b}\left( 1 \right){\text{ }} = {\text{ }}0$ Identity Rule –When the value of the base b and the argument of the logarithm (inside the parenthesis) are equal then,$lo{g_b}\left( b \right){\text{ }} = {\text{ }}1$ Log of Exponent Rule –If the base of the exponent is equal to the base of the log then the logarithm of the exponential number is equal to the exponent.$lo{g_b}\left( {{b^k}} \right){\text{ }} = {\text{ }}k$ Exponent of Log Rule –Raising the logarithm of a number to its base is equal to the number.${b^{logb\left( k \right)}} = k$

### Here are a Few Examples That Show How the Above Basic Rules Work

Example 1 –

Log 40, which can be further written as, $Log{\text{ }}\left( {20 \times {\text{ }}2} \right)$ by-product rule

$= {\text{ }}log{\text{ }}20{\text{ }} + {\text{ }}log{\text{ }}2$ which is equal to log 40

Example 2 –

Find the value of log4(4)?

logb(b) = 1, by identity rule

Therefore, log4(4) = 1.

### The Formula for Change of Base

The logarithm can be in the form of log base e or log base 10 or any other bases. Here’s the general formula for change of base -

${\mathbf{Lo}}{{\mathbf{g}}_{\mathbf{b}}}\;{\mathbf{x}}{\text{ }} = {\text{ }}\frac{{{\mathbf{Lo}}{{\mathbf{g}}_{\mathbf{a}}}\;{\mathbf{x}}}}{{{\mathbf{Lo}}{{\mathbf{g}}_{\mathbf{a}}}\;{\mathbf{b}}}}\;$

To find the value of log base 2, we first need to convert it into log base 10 which is also known as a common logarithm.

$Log{\text{ }}base{\text{ }}2{\text{ }}of{\text{ }}x = \frac{{ln\left( x \right)}}{{ln\left( 2 \right)}}$

### Now you Might Think what Common Logarithmic Function is?

• Common Logarithmic Function or Common logarithm is the logarithm with base equal to 10.

• It is also known as the decimal logarithm because of its base.

• The common logarithm of x is denoted as log x.

• Example: log 100 = 2 (Since 102= 100).

### How to Calculate Log Base 2?

This is how to find log base 2 -

• According to the log rule,

Log Rule -

$log_{b}(x) = y$

$b^{y} = x$

•  Suppose we have a question, log216 = x

• Using the log rule,

• $\;{2^x} = {\text{ }}16$

• We know that 16 in powers of 2 can be written as $\left( {2 \times 2 \times 2 \times 2{\text{ }} = 16} \right){\text{ }},{2^x} = {2^4}$

• Therefore, x is equal to 4.

### Questions to be Solved –

Question 1) Calculate the value of log base 2 of 64.

Solution) Here,

X= 64

Using the formula, $Log{\text{ }}base{\text{ }}2{\text{ }}of{\text{ }}X = \frac{{ln\left( {64} \right)}}{{ln\left( 2 \right)}} = 6$.

Log base 2 of 64 =$\frac{{ln\left( {64} \right)}}{{ln\left( 2 \right)}} = 6$.

Therefore, Log base 2 of 64 = 6

Question 2) Find the value of log2(2).

Solution) To find the value of log2(2) we will use the basic identity rule,

$lo{g_b}\left( b \right){\text{ }} = {\text{ }}1,$.

Therefore, log2(2) = 1.

Question 3) What is the value of log 2 base 10?

Solution) The value of log 2 base 10 can be calculated by the rule,

$Lo{g_a}\left( b \right){\text{ }} = \frac{{\log b}}{{\log a}}$.

$Lo{g_{10}}\left( 2 \right){\text{ }} = \frac{{\log 2}}{{\log 10}}\; = {\text{ }}0.3010$.

Therefore, the value of log 2 base 10 = 0.3010.

Question 4) What is the value of log 10 base 2?

Solution) The value of log 10 base 2 can be calculated by the rule,

$Lo{g_b}\left( a \right){\text{ }} = \frac{{\log b}}{{\log a}}$.

$Lo{g_2}\left( {10} \right){\text{ }} = \frac{{\log 10}}{{\log 2}}\; = {\text{ 3}}{\text{.3 = 2}}$.

Therefore, the value of log 10 base 2 = 3.32.

1. What do you mean by log 2?

In mathematics log 2 is equal to log (2, x). It represents the logarithmic value of x with base equal to 2.

2. How to find log base 2 on a calculator?

Calculating logarithm with base 2 is an easy task on a calculator. For example, you want to calculate the logarithm base 2 of 8, then these steps show how to find log base 2– 3. What is the value of log 2 Base 2?

The value of log 2 base 2 is equal to 1.

4. How to calculate log base 2 without a calculator?

According to the log rule, this is how to calculate log base 2.

According to the log rule, this is how to calculate log base 2. Suppose we have a question, log216 = x

Using the log rule,

2x = 16

We know that 16 in powers of 2 can be written as (2×2×2×2 =16)

2x =24

Therefore, x is equal to 4.